64 



TBANSPOBTATION OF DEBRIS BY BUNNING WATER. 



Curves (7a) and (11 a) reach the origin of 

 coordinates that is, their equations indicate 

 that at the zero of slope there is no capacity 

 for traction. Formula (11 a) gives small but 

 finite capacities for very low slopes; but under 

 formula (7a) finite capacities cease when the 

 slope falls to 0.29 per cent, and for lower slopes 

 there is indication of negative capacities. If 

 the conditions of traction permitted, negative 

 capacity might be interpreted as capacity for 

 traction upstream ; but as this is inadmissible, 

 the negative values may be classed as surd 

 results arising from the imperfection of the 

 correlation between an abstract formula and a 

 concrete problem. The curves of (8a) and (9a) 

 also intersect the axis of slope at some distance 

 from the origin, and their extensions indicate 

 negative capacity. The curve of (lOa) becomes 

 tangent to the axis of slope at the point corre- 

 sponding to a slope of 0.39 per cent and there 

 ends, having no continuation below the axis. 

 It is the real limb of a parabola of which all 

 other parts are imaginary. It expresses to the 

 eye the implication of formula (lOa) that trac- 

 tion ceases when the slope is reduced to 0.39 

 per cent, and that its cessation is not abrupt 

 but gradual; and also the implication of the 

 general formula (10) that traction ceases when 

 the slope is reduced to the value a. 



It is a matter of observation that when slope 

 is gradually reduced, the current becoming 

 feebler and the capacity gradually less, the 

 zero of capacity is reached before the zero of 

 slope. For each group of conditions (fineness, 

 width, discharge) there is a particular slope 

 corresponding to the zero of capacity. It is 

 also a matter of observation that the change 

 in capacity near the zero is gradual. Formulas 

 (7a), (8a), (9a), and (lOa) therefore accord 

 with the data of observation in the fact that 

 they connect the zero of capacity with a finite 

 slope; formula (Ha), which connects zero 

 capacity with zero slope, is discordant. Also, 

 formulas (lOa) and (lla) accord with the data 

 of observation in that they make the approach 

 of capacity to its zero gradual; while formulas 

 (7a), (8a), and (9a), which make the arrival of 

 capacity at its zero abrupt, are in that respect 

 discordant. 



But one of the formulas (lOa), shows quali- 

 tative agreement with both of the criteria 

 applied through extrapolation to low slopes; 

 and that formula is one of the two which 



respond best to the criterion applied through 

 extrapolation to high slopes. That type of 

 formula, or 



=&,(-)"--- ..(10) 



was therefore selected for the reduction of the 

 more or less irregular series of observational 

 values of capacity to orderly series better 

 suited for comparative study. 



In rewriting the formula the coefficient is 

 changed from 6 to 6 U because corresponding 

 coefficients 6 2 , 6 3 , etc., are to be used in a series 

 of formulas expressing the relations of capacity 

 to various conditions. As slope is a ratio 

 between lengths, (S a) n is of zero dimensions 

 and &! is of the unit C; it is the value of capacity 

 when S a=l. 



The slope which is barely sufficient to 

 initiate traction has been defined (p. 35) as 

 the competent slope. To whatever extent a 

 represents the competent slope the formula 

 has a rational basis. The local potential 

 energy of a stream, or the energy available at 

 any cross section in a unit of time, is simply 

 proportional to the product of discharge by 

 slope or, if the discharge be constant, is pro- 

 portional to the slope. So long as the slope 

 is less than that of competence the energy is 

 expended on resistances at contact with wetted 

 perimeter and air and on internal work 

 occasioned by those resistances. When the 

 slope exceeds the competent slope, part of the 

 energy is used as before and part is used in 

 traction. The change from competent slope 

 to a steeper slope increases the available 

 energy by an amount proportional to the 

 increase of slope, and the increase of energy is 

 associated with the added work of traction. 

 Capacity for traction, beginning at competent 

 slope, increases pari passu with the increase 

 of the excess of slope above the competent 

 slope, and there is manifest propriety in treat- 

 ing it as a function of the excess of slope 

 rather than of the total slope. It is of course 

 also a function of the total slope; but an 

 adequate formula for its relation to the excess 

 of slope may reasonably be supposed to be 

 simpler than a formula for its relation to total 

 slope. If a represents competent slope, then 

 the relation of capacity to S a should be 

 simpler than its relation to 8. 



Instructive information as to the relative 

 simplicity of the two functions is obtained by 



